On an example by Poincaré and sums with Kronecker sequence
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913536181010432 |
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| author | Moshchevitin, Nikolay |
| author_facet | Moshchevitin, Nikolay |
| contents | This short and simple communication is motivated by recent papers by L. Colzani and A. Kochergin. We give a brief analysis of an example by Poincaré related to sums of the type $ \sum_{k=0}^{t-1} f(kα+{x})$ where $f$ is a continuous periodic function and $α$ is irrationaland its recent generalisations. Most of the constructions under consideration are well-known. In this note, we just wanted to bring all the results together and give a general and improved multi-dimensional formulation of a recent result by A. Kochergin, prove non-existence of a universal continuous function and discuss some of the related results in terms of Diophantine Approximation. In particular, in our opinion smoothness results involving Diophantine exponents $ω$, $\hatω$ and $\hatλ$ had never been documented before. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_05003 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On an example by Poincaré and sums with Kronecker sequence Moshchevitin, Nikolay Number Theory 11K31, 11J13 This short and simple communication is motivated by recent papers by L. Colzani and A. Kochergin. We give a brief analysis of an example by Poincaré related to sums of the type $ \sum_{k=0}^{t-1} f(kα+{x})$ where $f$ is a continuous periodic function and $α$ is irrationaland its recent generalisations. Most of the constructions under consideration are well-known. In this note, we just wanted to bring all the results together and give a general and improved multi-dimensional formulation of a recent result by A. Kochergin, prove non-existence of a universal continuous function and discuss some of the related results in terms of Diophantine Approximation. In particular, in our opinion smoothness results involving Diophantine exponents $ω$, $\hatω$ and $\hatλ$ had never been documented before. |
| title | On an example by Poincaré and sums with Kronecker sequence |
| topic | Number Theory 11K31, 11J13 |
| url | https://arxiv.org/abs/2310.05003 |